Abstract
The article considers a technique for proving the Hodge, Tate, and Mumford-Tate conjectures for a simple complex Abelian variety of non-exceptional dimension under the condition that , where is the skew field of classical quaternions. The simple -dimensional Abelian varieties over a number field ( is a prime, ) are studied in detail. An application is given of Minkowski's theorem on unramified extensions of the field to the arithmetic and geometry of certain Abelian varieties over the field of rational numbers.
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